What is Mollifiers? — Mathematics

Definition 12.8 University Mathematics — Year 3 · Chapter 12 — The Lp Spaces

The function

\rho(x) = \begin{cases} c\,\exp\Bigl(-\dfrac{1}{1 - \norm x^2}\Bigr) & \norm x < 1,\\ 0 & \norm x \geq 1, \end{cases}

with $c$ normalizing $\int\rho = 1$ , is $\mathcal C^\infty$ on $\R^d$ : the point is that $t \mapsto \eu^{-1/t}\mathbf 1_{t>0}$ is $\mathcal C^\infty$ on $\R$ , all its derivatives at $0^+$ being $0$ (each derivative is $P(1/t)\eu^{-1/t}$ for a polynomial $P$ , which tends to $0$ ; induction). For $\varepsilon > 0$ set $\rho_\varepsilon(x) = \varepsilon^{-d}\rho(x/\varepsilon)$ : supported in $\bar B(0, \varepsilon)$ , still of integral $1$ .

Examples

Example 12.10 (Mollifying $\abs x$ , with rates)

Take $f(x) = \abs x$ on $\R$ (locally $L^1$ ; the theorem applies on every bounded window) and a symmetric mollifier $\rho_\varepsilon$ . Then

f_\varepsilon(x) = (f * \rho_\varepsilon)(x) = \int\abs{x - y}\,\rho_\varepsilon(y)\,\dd y

is $\mathcal C^\infty$ ; away from the kink, nothing happens: for $\abs x \geq \varepsilon$ , $\abs{x - y}$ is linear in $x$ on the support of $\rho_\varepsilon$ , so $f_\varepsilon(x) = \abs x$ exactly (symmetry kills the correction). Near $0$ , smoothing costs precisely

0 \leq f_\varepsilon(0) = \int\abs y\,\rho_\varepsilon(y)\,\dd y \leq \varepsilon, \qquad \norm{f_\varepsilon - f}_\infty \leq \varepsilon :

the approximation error is confined to the $\varepsilon$ -neighborhood of the singularity and is of its size. Meanwhile $f_\varepsilon'' \geq 0$ everywhere ( $f$ is convex, and convolution against $\rho_\varepsilon \geq 0$ preserves convexity), with $\int f_\varepsilon'' = f_\varepsilon'(\infty) - f_\varepsilon'(-\infty) = 2$ : the second derivative is a bump of mass $2$ squeezed into width $O(\varepsilon)$ , so $\norm{f_\varepsilon''}_\infty \gtrsim \varepsilon^{-1}$ . Smoothing is a trade: uniform error $O(\varepsilon)$ against derivative blow-up $O(\varepsilon^{-1})$ — the exact exchange rate that quantitative analysis (interpolation inequalities, Problem 12.1’s circle of ideas) formalizes.

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